Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(183.682394985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 2 x^{7} - 20109 x^{6} - 684862 x^{5} + 88807787 x^{4} + 5739442252 x^{3} + 94265489852 x^{2} + \cdots - 3335964735712 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3^{4}\cdot 7^{6} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 2 x^{7} - 20109 x^{6} - 684862 x^{5} + 88807787 x^{4} + 5739442252 x^{3} + 94265489852 x^{2} + \cdots - 3335964735712 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 120627551295437 \nu^{7} + \cdots - 42\!\cdots\!56 ) / 29\!\cdots\!32 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 97\!\cdots\!05 \nu^{7} + \cdots - 22\!\cdots\!08 ) / 34\!\cdots\!88 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 74\!\cdots\!93 \nu^{7} + \cdots + 39\!\cdots\!08 ) / 19\!\cdots\!46 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 58\!\cdots\!65 \nu^{7} + \cdots - 17\!\cdots\!16 ) / 34\!\cdots\!88 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!57 \nu^{7} + \cdots - 10\!\cdots\!72 ) / 34\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 79\!\cdots\!91 \nu^{7} + \cdots - 19\!\cdots\!28 ) / 17\!\cdots\!94 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21\!\cdots\!89 \nu^{7} + \cdots + 39\!\cdots\!08 ) / 16\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( -7\beta_{6} + 7\beta_{5} - 7\beta_{4} - 28\beta_{3} - 49\beta_{2} - 9\beta _1 + 588 ) / 2352 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 70 \beta_{7} - 245 \beta_{6} + 343 \beta_{5} + 623 \beta_{4} - 3682 \beta_{3} - 9807 \beta_{2} + \cdots + 11825268 ) / 2352 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1064 \beta_{7} - 36092 \beta_{6} + 53739 \beta_{5} - 18074 \beta_{4} - 195895 \beta_{3} + \cdots + 319761456 ) / 1176 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 454244 \beta_{7} - 5318033 \beta_{6} + 14544033 \beta_{5} + 5771059 \beta_{4} + \cdots + 135038101380 ) / 2352 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41128038 \beta_{7} - 917399189 \beta_{6} + 2184929775 \beta_{5} - 62503651 \beta_{4} + \cdots + 12739720043988 ) / 2352 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 12992325 \beta_{7} - 6763767820 \beta_{6} + 22081095159 \beta_{5} + 4797486036 \beta_{4} + \cdots + 139936468813680 ) / 168 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 743501342906 \beta_{7} - 13786689240695 \beta_{6} + 42416034831499 \beta_{5} + 3071612229231 \beta_{4} + \cdots + 22\!\cdots\!20 ) / 2352 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 27.0000 | 0 | −451.992 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 27.0000 | 0 | −366.561 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 27.0000 | 0 | −100.510 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 27.0000 | 0 | −79.3703 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 27.0000 | 0 | −70.2340 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 27.0000 | 0 | 189.617 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 27.0000 | 0 | 396.248 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 27.0000 | 0 | 482.802 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 588.8.a.n | yes | 8 |
| 7.b | odd | 2 | 1 | 588.8.a.m | ✓ | 8 | |
| 7.c | even | 3 | 2 | 588.8.i.q | 16 | ||
| 7.d | odd | 6 | 2 | 588.8.i.r | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 588.8.a.m | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 588.8.a.n | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 588.8.i.q | 16 | 7.c | even | 3 | 2 | ||
| 588.8.i.r | 16 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 393032 T_{5}^{6} - 12705984 T_{5}^{5} + 42177514488 T_{5}^{4} + 2844112124160 T_{5}^{3} + \cdots - 33\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T - 27)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 33\!\cdots\!00 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 25\!\cdots\!24 \)
|
| $13$ |
\( T^{8} + \cdots + 76\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots - 10\!\cdots\!04 \)
|
| $19$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
|
| $23$ |
\( T^{8} + \cdots + 19\!\cdots\!24 \)
|
| $29$ |
\( T^{8} + \cdots - 16\!\cdots\!96 \)
|
| $31$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $37$ |
\( T^{8} + \cdots + 43\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots - 43\!\cdots\!36 \)
|
| $43$ |
\( T^{8} + \cdots + 68\!\cdots\!44 \)
|
| $47$ |
\( T^{8} + \cdots - 12\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots - 12\!\cdots\!64 \)
|
| $59$ |
\( T^{8} + \cdots + 40\!\cdots\!24 \)
|
| $61$ |
\( T^{8} + \cdots - 12\!\cdots\!44 \)
|
| $67$ |
\( T^{8} + \cdots - 97\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots + 51\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots + 31\!\cdots\!76 \)
|
| $83$ |
\( T^{8} + \cdots - 15\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 14\!\cdots\!44 \)
|
| $97$ |
\( T^{8} + \cdots - 18\!\cdots\!64 \)
|
| show more | |
| show less |